The most useful form of Maxwell's equations in solving a wide
class of electromagnetic problems is the differential or ``point''
form, along with the associated boundary conditions. Maxwell's
equations in time-varying, differential form can be written as,
\begin{subequations}
\begin{align}
    \nabla\cdot\vec{\mathcal{D}}&=\rho_{ev}&&\text{Gauss's Electric Field Law}\label{eqn:gefl}\\
    \nabla\cdot\vec{\mathcal{B}}&=\rho_{mv}=0&&\text{Gauss's Magnetic Field Law}\label{eqn:gmfl}\\
    \nabla\times\;\vec{\mathcal{E}}&=-\frac{\partial\vec{\mathcal{B}}}{\partial{t}}-\vec{\mathcal{M}}&&\text{Faraday's Law}&&\label{eqn:fl}\\
    \nabla\times\,\vec{\mathcal{H}}&=\frac{\partial\vec{\mathcal{D}}}{\partial{t}}+\vec{\mathcal{J}}&&\text{Ampere's Law}\label{eqn:al}
\end{align}
\end{subequations}
The script variables in (\ref{eqn:gefl})--(\ref{eqn:al}) defined
in table \ref{tbl:emvarfs} are real functions of the spatial
coordinate $\vec{r}$ and the time coordinate $t$. The MKS system
of units is used to describe all units of measure.
\begin{table}[!hbp]
\caption{Electromagnetic Variables}\label{tbl:emvarfs}
\begin{center}
\begin{tabular}{|c|l|c|}
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  \hline
  Symbol & Variable Definition & Units\\
  \hline
  $\vec{\mathcal{H}}$ & magnetic field intensity & ${A}/{m}$\\
  $\vec{\mathcal{E}}$ & electric field intensity & ${V}/{m}$\\
  $\vec{\mathcal{B}}$ & magnetic flux density & ${Wb}/{m^2}$\\
  $\vec{\mathcal{D}}$ & electric flux density & ${C}/{m^2}$\\
  $\rho_{mv}$ & magnetic volumetric charge density & $Wb/m^3$\\
  $\rho_{ev}$ & electric volumetric charge density & $C/m^3$\\
  $\vec{\mathcal{M}}$ & magnetic current density (source and/or sink) & $V/m^2$\\
  $\vec{\mathcal{J}}$ & electric current density (source and/or sink) & $A/m^2$\\
  \hline
\end{tabular}
\end{center}
\end{table}

The electric current density $\vec{\mathcal{J}}$ and magnetic
current density $\vec{\mathcal{M}}$ can be further broken down
into the superposition of a source (impressed) current density
$(\vec{\mathcal{J}_i},\vec{\mathcal{M}_i})$ and a sink (lossy)
current density $(\vec{\mathcal{J}_\ell},\vec{\mathcal{M}_\ell})$
as follows:
\begin{align}
    \vec{\mathcal{J}}&=\vec{\mathcal{J}_i}+\vec{\mathcal{J}_\ell}\label{eqn:Js}\\
    \vec{\mathcal{M}}&=\vec{\mathcal{M}_i}+\vec{\mathcal{M}_\ell}\label{eqn:Ms}
\end{align}
The source current density
$(\vec{\mathcal{J}_i},\vec{\mathcal{M}_i})$ is defined as a
current source that adds power to a given system and a sink
current density $(\vec{\mathcal{J}_\ell},\vec{\mathcal{M}_\ell})$
is defined as one that dissipates power from the system. This will
be explained in more detail in section \ref{sec:PowerandEnergy} on
page \pageref{sec:PowerandEnergy}. In \cite{Balanis:1989}, the
electric sink current density is also referred to as a conduction
current. The source of the electric field is the electric charge,
or the electric charge density $\rho_{ev}$. Current is the flow of
charge, therefore the electric current density
$\vec{\mathcal{J}_i}$ is also a source of the electric field.
Similarly, the sources of the magnetic field is the fictitious
magnetic charge density $\rho_{ev}$ and the the fictitious
magnetic current density $\vec{\mathcal{M}_i}$. In reality, there
is no such thing as a magnetic charge or magnetic current density.
The real source of the magnetic field is a loop of electric
current or magnetic dipole. These magnetic sources are nothing
more than a mathematical convenience to balance Maxwell's
equations and are very useful for describing fields in terms of
equivalent sources.

By taking the divergence of (\ref{eqn:gefl}) and knowing the
divergence of the curl of a vector is equal to zero yields,
\begin{equation}\label{eqn:continuityeqn}
    \nabla\cdot\vec{\mathcal{J}}=-\frac{\partial{\rho_{ev}}}{\partial{t}}\qquad\text{Continuity Equation}
\end{equation}
This is known as the \emph{continuity equation} and shows the
charge density and the current density are related. The
\emph{continuity equation} states that charge is conserved.
Maxwell used the \emph{continuity equation} to introduce his
displacement current term into Ampere's law.
